# -*- coding: utf-8 -*-

"""64. 最小路径和
给定一个包含非负整数的 m x n 网格 grid ，请找出一条从左上角到右下角的路径，使得路径上的数字总和为最小。
说明：每次只能向下或者向右移动一步。

示例 1：
输入：grid = [[1,3,1],[1,5,1],[4,2,1]]
输出：7
解释：因为路径 1→3→1→1→1 的总和最小。

示例 2：
输入：grid = [[1,2,3],[4,5,6]]
输出：12

提示：
m == grid.length
n == grid[i].length
1 <= m, n <= 200
0 <= grid[i][j] <= 200"""

class Solution:
    """
    递归基础：
        f(0, x) = 首行前缀和
        f(x, 0) = 首行前缀和
    递归定义：
        f(i, j) = min(f(i-1, j), f(j-1, i)) + grid[i][j]

    按动态规划，将递归运算，转化为递推的状态缓存。可以就在原先的矩阵上更新。

    """
    def minPathSum(self, grid: list) -> int:
        row_count, col_count = len(grid), len(grid[0])
        j = 1
        while j < col_count:
            grid[0][j] += grid[0][j-1]
            j += 1
        i = 1
        while i < row_count:
            grid[i][0] += grid[i-1][0]
            i += 1

        i = 1
        while i < row_count:
            j = 1
            while j < col_count:
                grid[i][j] = min( grid[i][j-1], grid[i-1][j] ) + grid[i][j]
                j += 1
            i += 1

        return grid[row_count-1][col_count-1]

if __name__ == '__main__':
    print(Solution().minPathSum([[1,2,3],[4,5,6]]))
